(a) imagining there's one "other side" is a fallacy. There are multiple interest groups. Some more aligned with the President and his administration, some less.
(b) you'll have to be more specific about what you mean, but the "tantrum" I've heard is that the system as set up over-represents land over people, not that the result is illegitimate. A legitimate result in a badly-crafted system is materially different from claiming the process as designed is compromised.
I'm observing from the outside (read: the other side of the Atlantic), but my understanding is that people who say #NotMyPresident do it because of either a) a perceived lack of shared values between them and the president or b) in reference to him losing the popular vote, like several other presidents before him.
1. Questioning whether the Electoral College (and its tendency to devalue votes in some states) has a place in the modern US.
2. Questioning whether the election itself is completely rigged (via fraudulent votes).
#1 is the question many liberals have been asking. #2 is thee claim that the entire GOP has been making for years, despite their own investigations never turning up more than a few individuals voting fraudulently (but never systematic fraud perpetrated by the political left, as they claim).